### Interactions between Multiuser Diversity and Spatial Diversity Techniques in an Interference-limited

### Environment

Wan Choi* ^{∗}*, Nageen Himayat

*, Shilpa Talwar*

^{†}*, J.Y. Kim*

^{†}*, Albert Koo*

^{‡}*, Jane Choi*

^{‡}*, Yujin Noh*

^{‡}*, and Josep Kim*

^{‡}

^{‡}*∗*School of Engineering, Information and Communications University (ICU)
119 Munjiro, Yuseong-gu, Daejeon, 305-732, Korea

Email: w.choi@ieee.org

*†*Radio Communication Lab./ Communication Technology Lab.

Corporate Technology Group
Intel Corporation, Santa Clara, CA
Email:*{nageen.himayat, shilpa.talwar}@intel.com*

*‡*Intel Korea R&D Center (iKRC)
Intel Korea

**Abstract— This paper investigates the interaction between mul-****tiuser diversity and spatial diversity in an interference-limited en-****vironment based on post-receiver-processing signal-to-interference-**
**plus-noise ratio (SINR) distributions. If opportunistic scheduling is**
**employed, spatial diversity effects limit the achievable multiuser**
**diversity gain. This paper quantifies the interaction by using order**
**statistic theory and shows a spatial diversity technique with a larger**
**SINR variance can be more effective under opportunistic scheduling.**

**Through analysis and simulations, we show that the cyclic delay**
**diversity technique gets the most benefit from the opportunistic**
**scheduling among likely spatial diversity techniques and outperforms**
**space time block coding (STBC), even though STBC is generally**
**considered the most effective transmit diversity technique in a noise-**
**limited environment.**

I. INTRODUCTION

Multiple-input multiple output (MIMO) techniques have promised significant performance and capacity gains (refer to [1]–

[4] and the reference therein) in wireless communications. In ad- dition to MIMO techniques, multiuser diversity is considered an- other strong tool to increase spectral efficiency in future wireless communication systems. The idea itself was first proposed in [5]

and commercial adoption can be already found in several systems such as 1xEV-DO [6], [7]. The capacity increasing capability of multiuser diversity comes from exploiting the inherent channel variations across users in a system as diversity sources. The total throughput can be significantly improved by serving only the user(s) with the highest instantaneous signal-to-interference-plus- noise ratio (SINR) at a given time.

There are a large number of studies on exploiting multiuser diversity in multiple antenna settings [8]–[15]. However, there are few studies investigating the interaction between spatial diversity and multiuser diversity. In general, it has been just conjectured that there is a fundamental conflict between multiuser diversity and spatial diversity – multiuser diversity is owed to channel vari- ations but spatial diversity generally reduces channel variations of each user. The impact of multiuser diversity on the performance of space-time block codes (STBC) was investigated in [16]. The authors showed that multiuser diversity with no spatial diversity outperforms the schemes that employ both multiuser diversity and

spatial diversity in terms of effective signal-to-noise ratio (SNR) and spectral efficiency. The combined use of spatial diversity and multiuser diversity is studied more generally for various spatial diversity schemes including STBC and coherent beamforming by Larsson [17]. The paper concluded that, in general, spatial diversity with knowledge of channel state information (CSI) increases the total diversity gain in a system exploiting multiuser diversity, although the gain offered by diversity transmission is not as large as in a single user system.

Considering that future cellular systems will operate in increas- ingly interference-limited environments and are likely to adopt opportunistic transmission scheduling on top of various spatial diversity techniques, the effects of co-channel interference (CCI) and multiuser diversity on various spatial diversity techniques should be carefully studied. The paper [18] investigated the effects of CCI on spatial diversity techniques, but the focus was on SINR analysis so the interaction was not handled in that paper.

In this context, this paper analyzes the interactions between multiuser diversity and spatial diversity in an interference- limited environment. We compare the key characteristics of post- processing signal to interference plus noise ratio (SINR) distri- butions before and after opportunistic scheduling, and quantify the interaction between multiuser diversity and spatial diversity.

Our results show that even though STBC gets more diversity benefit (smaller variance) without opportunistic scheduling, the small SINR variance significantly limits the multiuser diversity gain so it has rather worse performance in terms of both spec- tral efficiency and error probability than other diversity tech- niques if opportunistic scheduling is adopted. The investigation is also extended to orthogonal frequency division multiple access (OFDMA) systems by system level simulations.

II. SPATIALDIVERSITYWITHOUTOPPORTUNISTIC

SCHEDULING

The effects of CCI on spatial diversity were investigated in [18].

In this section, we first summarize the results of [18] to investigate the effects of opportunistic scheduling in an interference-limited environment.

**1525-3511/07/$25.00 ©2007 IEEE **

*A. Analytical Assumptions and Model*

We list up the analytical assumptions in [18]. This paper is
also based on these assumptions: (1) An interference-limited
environment – thermal noise is neglected and the desired signal
power is*P*0. (2)*K independent and identically distributed (i.i.d.)*
co-channel interference with identical power*P**I*. (3) Desired and
interfering signals employ the same transmission scheme over the
same antenna configuration. – each transmitter has *M**t* transmit
antennas and each receiver unit has *M**r* receive antennas. (4)
Complex Gaussian fading channels are considered – both the
desired and interfering signals experience spatially white complex
Gaussian fading channels. Each channel coefficient of*M**t**× M**r*

MIMO channel matrices follows*CN (0, 1). (5) Channels are static*
during a symbol period. (6) Matched filter (MF) receivers without
knowledge of interfering signals are used.

*B. Post-processing SINR Distributions*

*1) SISO: Probability density function (PDF) of SISO is*
*f**γ**(γ) =* *P**I*

*P*0

*Γ(1 + K)*
*Γ(1)Γ(K)*

1
1 + ^{P}_{P}^{I}

0*γ*

_{1+K}*,* *γ ≥ 0* (1)

with

*E[γ] =* *P*0

*P**I**(K − 1)* *and Var[γ] =* *P*_{0}^{2}
*P*_{I}^{2}

*K*

*(K − 1)*^{2}*(K − 2).*
where *K > 1 and K > 2 for mean and variance, respectively.*

*2) Receive Diversity (RD) – MRC: PDF is given by*
*f**γ**(γ) =* *P**I*

*P*0

*Γ(M**r**+ K)*
*Γ(M**r**)Γ(K)*

*(P**I**γ/P*0)^{M}^{r}^{−1}

*(1 + P**I**γ/P*0)^{M}^{r}^{+K}*,* *γ ≥ 0* (2)
with

*E[γ] =* *P*0*M**r*

*P**I**(K − 1)* *and Var[γ] =* *P*_{0}^{2}
*P*_{I}^{2}

*M**r**(M**r**+ K − 1)*
*(K − 1)*^{2}*(K − 2).*
where *K > 1 and K > 2 for mean and variance, respectively.*

*3) Space Time Block Coding (STBC): PDF of STBC is*
*f**γ**(γ) =* *P**I*

*P*0

*Γ(M**r**M**t**+ KM**t*)
*Γ(M**r**M**t**)Γ(KM**t*)

*(P**I**γ/P*0)^{M}^{r}^{M}^{t}^{−1}

*(1 + P**I**γ/P*0)^{M}^{r}^{M}^{t}^{+KM}* ^{t}* (3)
with

*E[γ] =* *P*0*M**r**M**t*

*P**I**(KM**t**− 1)* and
*Var[γ] =* *P*_{0}^{2}

*P*_{I}^{2}

*M**r**M**t**(M**r**M**t**+ KM**t**− 1)*
*(KM**t**− 1)*^{2}*(KM**t**− 2)* *.*

where *KM**t* *> 1 and KM**t* *> 2 for mean and variance,*
respectively.

*4) Cyclic Delay Diversity (CDD) or Phase Shifted Transmis-*
*sion: PDF is given by*

*f**γ**(γ) =* *P**I*

*P*0

*Γ(M**r**+ K)*
*Γ(M**r**)Γ(K)*

*(P**I**γ/P*0)^{M}^{r}^{−1}

*(1 + P**I**γ/P*0)^{M}^{r}^{+K}*,* *γ ≥ 0* (4)
with

*E[γ] =* *P*0*M**r*

*P**I**(K − 1)* *and Var[γ] =* *P*_{0}^{2}
*P*_{I}^{2}

*M**r**(M**r**+ K − 1)*
*(K − 1)*^{2}*(K − 2).*
where *K > 1 and K > 2 for mean and variance, respectively.*

*5) Coherent Beamforming – Transmit MRC: PDF of coherent*
beamforming is given by

*f**γ**(γ) =* *P**I*

*P*0

*Γ(M**t**+ K)*
*Γ(M**t**)Γ(K)*

*(P**I**γ/P*0)^{M}^{t}^{−1}

*(1 + P**I**γ/P*0)^{M}^{t}^{+K}*,* *γ ≥ 0* (5)
with

*E[γ] =* *P*0*M**t*

*P**I**(K − 1)* *and Var[γ] =* *P*_{0}^{2}
*P*_{I}^{2}

*M**t**(M**t**+ K − 1)*
*(K − 1)*^{2}*(K − 2).*
where *K > 1 and K > 2 for mean and variance, respectively.*

The results are summarized in Table I. According to these
results, STBC is more sensitive to the number of interferers in
terms of the mean and the variance since the number of effective
interferers becomes *KM**t* as shown in the denominator. When
there are three independent and identically distributed (i.i.d.)
interferers with the same power*P**I*, cumulative density functions
(CDF) of spatial diversity techniques are given in Fig. 1. The
power ratio between the desired signal power*P*0and*P**I* is 7*.5dB.*

III. THEEFFECTS OFMULTIUSERDIVERSITY ONSPATIAL

DIVERSITYTECHNIQUES

Without multiuser diversity, a smaller SINR variance typically results in better spatial diversity performance if the mean values of SINR are the same or similar. However, if opportunistic schedul- ing is adopted, a smaller SINR variance may limit multiuser diversity gain. This section studies this conjecture and quantifies the effects of max-SINR scheduling exploiting multiuser diversity on spatial diversity techniques in an interference-limited environ- ment.

If we adopt a max-SINR scheduler which selects the user
with the highest SINR to investigate the effects of multiuser
diversity, the post-processing SINR of the selected user is*γ*eﬀ =
max*u=1,··· , U**γ**u* where *γ**u* is the SINR of user *u and U is the*
total number of users. Each*γ**u*follows the derived distribution in
Section II for a given diversity technique. Using order statistics,
PDF of*γ*eﬀ is given by

*f**γ*eff*(γ*eﬀ*) = U f**γ**(γ*eﬀ*)F**γ**(γ*eﬀ)* ^{U−1}* (6)
where

*F*

*γ*(·) is the cumulative density function (CDF) of γ.

Fig. 2 shows the CDFs of the spatial diversity techniques when
the numbers of users are 30 (*U = 30). Three i.i.d. interferers*
with the same power *P**I* are considered and *P*0*/P**I* *= 7.5dB.*

Compared to CDF’s in Fig. 1 without opportunistic scheduling,
1*× 2 RD, 2 × 1 coherent BF, and 2 × 2 CDD outperform 2 × 2*
STBC in the whole CDF region. Even SISO and 2*× 1 CDD*
are better than 2*× 2 STBC at the high percentage CDF region.*

The technique with a higher variance takes more advantages
of opportunistic scheduling so the mean value after scheduling
(*E[γ*eﬀ]) becomes larger.

Even though it is difficult to obtain an exact closed-form
expression of the mean value after scheduling in general, an upper
bound on*γ*eﬀ can be obtained by using order statistic theory on
an upper bound on the maximum random variable [19] as

E

*1≤u≤U*max *γ**u*

*≤ E[γ] +√U − 1*
*2U − 1*

*Var[γ].* (7)

This theoretical upper bound effectively quantifies the interaction
between spatial diversity and multiuser diversity. The mean value
after scheduling (*E[γ*eﬀ]) increases with not only the number of

users but also the SINR variance of each user Var[*γ]. As a result,*
the improvement of the mean value after scheduling (E[γeﬀ]) is
pronounced for the schemes with higher variances as confirmed in
Table III. This table also shows that the theoretical upper bound
is reasonably tight with sufficient number of users.

The increased mean values by multiuser diversity are directly translated into performance improvement. Average symbol error rate (SER) for QPSK and spectral efficiency according to the av- erage received SINR are shown in Fig. 3 and Fig. 4, respectively.

The spectral efficiency is calculated by (*SE = log*_{2}*(1 + γ*eﬀ))
and the symbol error probability for QPSK is given by

*SER = N**e*E

*Q*

*d*^{2}_{min}*γ*eﬀ

2

(8)

where *N**e* and*d**min* are the average number and the minimum
distance of the nearest neighbors of the given signal constellation,
respectively. The number of users is 30 and three i.i.d. interferers
are considered. Owing to the increased mean values after oppor-
tunistic scheduling, 1*× 2 RD, 2 × 1 coherent BF, and 2 × 2*
CDD outperform 2*× 2 STBC in terms of both SER and spectral*
efficiency. Although SISO and 2*× 1 CDD have worse SER than*
1*× 2 STBC for a fixed modulation, they have slightly higher*
spectral efficiency even than 2*× 2 STBC if we relax the fixed*
modulation cosntraint.

To more effectively demonstrate the interaction between mul-
tiuser diversity and spatial diversity, spectral efficiencies of var-
ious spatial diversity techniques versus the number of users are
shown in Fig. 5 where three i.i.d. interferers with the same power
*P**I*are considered and*P*0*/P**I* = 5dB. Spectral efficiencies of 1×2
RD, 2*× 2 CDD, and 2 × 1 coherent BF grow faster than 2 × 2*
STBC owing to the larger SINR variances. Even SISO and 2*× 1*
CDD have higher spectral efficiencies than 2*× 2 STBC if the*
number of users is greater than 20.

For CDD, it should be also noted that as the number of users
increases there is more likely to be the user whose desired signals
are co-phased such that *h*11*+ e*^{−jθ}^{0}*h*12 = (*|h*11*| + |h*12*|)e** ^{−jφ}*.
These chances are neglected in analysis, but they contribute to
the performance enhancement of CDD in frequency selective user
scheduling.

IV. ORTHOGONALFREQUENCYDIVISIONMULTIPLEACCESS

(OFDMA) SYSTEMS

In this section, we investigate the interaction between multiuser diversity and spatial diversity in OFDMA systems over flat fading channels. In OFDMA systems, each resource block consisting of multiple sub-carriers over several OFDM symbols is allocated to users by a specific scheduling algorithm. An effective SINR characterizes the signal quality of a resource block for scheduling in OFDMA systems and there are several ways to calculate an effective SINR [20]–[26]. In this paper, we adopt the exponential- effective-SINR (EESM) in 3GPP LTE [24] and proportional fair scheduling [6], [9].

If channel quality indicator (CQI) feedback information from mobile units is available, adaptive modulation and coding selec- tion (MCS) and opportunistic scheduling can be adopted using the CQI feedback information. If a resource block consists of contiguous sub-carriers, then more fluctuations across resource

blocks result in more benefit from frequency selective oppor- tunistic scheduling. In this scenario, statistical characteristics of an effective SINR (EESM) become very similar to those of each SINR in a single carrier system since the sub-carriers within a resource block are highly correlated.

The performance of frequency selective proportional fair scheduling depends on EESM fluctuations across resource blocks.

For similar mean values of EESM, a larger EESM variance yields better performance and the scheme with lower SINR correla- tions gets more benefit from frequency selective opportunistic scheduling. The EESM fluctuations across resource blocks in the contiguous sub-carrier case can be well captured by SINR cross correlation between different sub-carriers.

*A. Cross Correlation Properties between Resource Blocks*
For analytical tractability, we only derive correlation between
desired signals on different resource blocks instead of correlation
between the exact effective SINRs on different resource blocks.

Considering that the desired signal is independent of interfer-
ing signals, the correlation property between desired signals on
different resource blocks effectively characterizes the frequency
diversity effect for a particular spatial diversity technique in
an OFDMA system. This section focuses on the three most
interesting spatial diversity techniques – 1*× 2 RD, 2 × 2 STBC,*
and 2*× 2 CDD.*

Let *E[|h*^{(i)}*mn**|*^{2}*|h*^{(j)}*pq**|*^{2}*] = ρ where h*^{(i)}*mn* denotes the channel
coefficient from the *nth transmit antenna to the mth receive*
antenna on the *ith resource block. The value of ρ equals to 1*
if *|h*^{(i)}*mn**|*^{2} and*|h*^{(j)}*pq**|*^{2} are uncorrelated and goes to 2 if they are
closely correlated. Note that *E[|h*^{(i)}*nm**|*^{2}*|h*^{(j)}*pq**|*^{2}] = 1 for different
*i and j if n = p or m = q since we assume that antennas are*
uncorrelated.

The cross correlation between the desired signals on different
resource blocks of 1*× 2 RD is given by*

*ρ*RD=

E

*|h*^{(i)}_{1} *|*^{2}+*|h*^{(i)}_{2} *|*^{2}

*|h*^{(j)}_{1} *|*^{2}+*|h*^{(j)}_{2} *|*^{2}

*− E*

*|h*^{(i)}_{1} *|*^{2}+*|h*^{(i)}_{2} *|*^{2}
E

*|h*^{(j)}_{1} *|*^{2}+*|h*^{(j)}_{2} *|*^{2}

*/*

Var

*|h*^{(i)}_{1} *|*^{2}+*|h*^{(i)}_{2} *|*^{2}
Var

*|h*^{(j)}_{1} *|*^{2}+*|h*^{(j)}_{2} *|*^{2}

*=ρ − 1* (9)

since*E[|h*^{(i)}*nm**|*^{4}] = 2 and*E[|h*^{(i)}*nm**|*^{2}*] = 1 for h*^{(i)}*nm**∼ CN (0, 1).*

For 2*× 2 STBC, the cross correlation between the desired*
signals on different resource blocks is

*ρ*STBC=

E

*|h*^{(i)}_{11}*|*^{2}+*|h*^{(i)}_{12}*|*^{2}+*|h*^{(i)}_{21}*|*^{2}+*|h*^{(i)}_{22}*|*^{2}

*·*

*|h*^{(j)}_{11}*|*^{2}+*|h*^{(j)}_{12}*|*^{2}+*|h*^{(j)}_{21}*|*^{2}+*|h*^{(j)}_{22}*|*^{2}

*−16*

*/*

Var

*|h*^{(i)}_{11}*|*^{2}+*|h*^{(i)}_{12}*|*^{2}+*|h*^{(i)}_{21}*|*^{2}+*|h*^{(i)}_{22}*|*^{2}

*/*

Var

*|h*^{(j)}_{11}*|*^{2}+*|h*^{(j)}_{12}*|*^{2}+*|h*^{(j)}_{21}*|*^{2}+*|h*^{(j)}_{22}*|*^{2}

*=ρ − 1.* (10)

Finally, the cross correlation between the desired signals on

different resource blocks of 2*× 2 CDD is given by*
*ρ*CDD=

E

*h*^{(i)}_{11} *+ e*^{−jφ}*h*^{(i)}_{12}^{2}+*h*^{(i)}_{21} *+ e*^{−jφ}*h*^{(i)}_{22}^{2}

*·*

*h*^{(j)}_{11} *+ e*^{−jθ}*h*^{(j)}_{12}^{2}+*h*^{(j)}_{21} *+ e*^{−jθ}*h*^{(j)}_{22}^{2}

*−16*

*/*

Var

*h*^{(i)}_{11} *+ e*^{−jφ}*h*^{(i)}_{12}^{2}+*h*^{(i)}_{21} *+ e*^{−jφ}*h*^{(i)}_{22}^{2}

*/*

Var

*|**h*^{(j)}_{11} *+ e*^{−jθ}*h*^{(j)}_{12}^{2}+*h*^{(j)}_{21} *+ e*^{−jθ}*h*^{(j)}_{22}^{2}

=1

2*(ρ − 1).* (11)

since*h*^{(i)}_{11}*+ e*^{−jφ}*h*^{(i)}_{12} *∼ CN (0, 2) so E*

*h*^{(i)}_{11}*+ e*^{−jφ}*h*^{(i)}_{12}^{2}

= 2 andE

*h*^{(i)}_{11}*+ e*^{−jφ}*h*^{(i)}_{12}^{4}

= 8.

The cross correlation properties of the spatial diversity tech-
niques show that 2*× 2 CDD has the lowest cross correlation*
between SINRs on different resource blocks among other spatial
diversity techniques. Owing to this reduced cross correlation of
post-processing SINRs across resource blocks, CDD effectively
exploits frequency diversity gain in multi-carrier systems. Along
with the SINR analysis in a single carrier system, the cross
correlation properties characterize the performance of spatial
diversity techniques in OFDMA systems well.

*B. Spectral Efficiencies*

This section presents system level simulation results based on the 3GPP contribution [27] and shows that our theoretical analysis can be consistently applied to predict the observed trends. Matched filter receivers are considered in the system level simulations and other simulation parameters and environments are the same as those of [27].

Fig. 6 shows the CDF of EESM in flat fading channels
across resource blocks before frequency selective opportunistic
scheduling. In the figure, the mean values of 1×2 RD, 2×2 STBC,
and 2×2 CDD are 6.3 dB, 6.6 dB, and 6.4 dB, respectively, while
the variances of 1*× 2 RD, 2 × 2 STBC, and 2 × 2 CDD are 46.1*
dB, 37*.9 dB, and 46.5 dB, respectively. Because the sub-carriers*
within a resource block are contiguous, the sub-carriers are
typically highly correlated and hence the statistical characteristics
of EESM of a specific resource block are similar to the post-
processing SINR in a single carrier system. Correspondingly, the
mean values of the spatial diversity techniques are similar and
STBC has the lowest variance as in a single carrier system.

The CDF of EESM of scheduled users in flat fading channels
is shown in Fig. 7. The distributions are quite changed after
opportunistic scheduling in an OFDMA system. The mean value
of 2*× 2 CDD after scheduling becomes 9.5 dB while the mean*
values of 1*× 2 RD and 2 × 2 STBC after scheduling are 6.5 dB*
and 6*.9 dB, respectively. Since the proportional fair scheduling*
exploits not only the variation of each EESM but also EESM
fluctuations across resource blocks, the mean value of 2 *× 2*
CDD after scheduling becomes the largest even though the SINR
variances of 1×2 RD and 2×2 CDD are almost the same before
proportional fair scheduling. These results confirm the theoretical
observation that CDD induces more fluctuations across EESMs as

shown in Subsection IV-A and hence takes the most benefits from
frequency selective user scheduling. The increased mean value of
CDD owing to frequency selective multiuser diversity directly
yields higher spectral efficiency. The average spectral efficiencies
of 1*× 2 RD, 2 × 2 STBC, and 2 × 2 CDD are 1.17bps/Hz/sector,*
*1.21bps/Hz/sector, and 1.56bps/Hz/sector, respectively.*

V. CONCLUSION

This paper has investigated the interaction between multiuser diversity and spatial diversity techniques in an interference- limited environment. An effective spatial diversity technique generally reduces SINR variance, but the reduced variation of SINR limits the multiuser diversity gain. We have quantified the interaction by using order statistic theory on an upper bound on the maximal random variable. The analytical and simulation results have shown that the cyclic delay diversity technique outperforms STBC in terms of both spectral efficiency and error probability and gets the most benefit under opportunistic schedul- ing even though STBC generally outperforms other transmit diversity techniques in a noise-limited environment.

REFERENCES

[1] G. Foschini and M. Gans, “On limits of wireless communications in a fading
*environment when using multiple antennas,” Wireless Personal Commun.,*
vol. 6, pp. 311–335, Mar. 1998.

*[2] E. Teletar, “Capacity of multi-antenna Gaussian channels,” European Trans.*

*Telecommun., vol. 6, pp. 585–595, Nov.-Dec. 1999.*

[3] A. Goldsmith, S. Jafar, N. Jindal, and S. Vishwanath, “Capacity limits of
*MIMO channels,” IEEE Journal on Sel. Areas in Communications, vol. 21,*
no. 5, pp. 684–702, June 2003.

[4] S. N. Diggavi, N. Al-Dhahir, A. Stamoulis, and A. Calderbank, “Great ex-
*pectations: The value of spatial diversity in wireless networks,” Proceedings*
*of the IEEE, vol. 92, no. 2, pp. 219–70, Feb. 2004.*

[5] R. Knopp and P. Humblet, “Information capacity and power control in
*single-cell multiuser communications,” in Proc., IEEE Intl. Conf. on Com-*
*munications, June 1995, pp. 331–335.*

*[6] CDMA 2000 (IS-856): High rate packet data air interface specification,*
TIA/EIA Std., 2000.

[7] P. Bender, P. Black, M. Grob, R. Padovani, N. Sinduhshayana, and A. Viterbi,

“CDMA/HDR: A bandwidth efficient high speed wireless data service for
*nomadic usres,” IEEE Communications Magazine, vol. 38, no. 7, pp. 70–77,*
July 2000.

[8] R. W. Heath, Jr., M. Airy, and A. J. Paulraj, “Multiuser diversity for MIMO
*wireless systems with linear receivers,” in Proc., IEEE Asilomar, Pacific*
Grove, California, Nov. 2001, pp. 1194 –1199.

[9] P. Viswanath, D. N. C. Tse, and R. Laroia, “Opportunistic beamforming
*using dumb antennas,” IEEE Trans. on Info. Theory, vol. 48, no. 6, pp.*

1277–1294, June 2002.

[10] W. Rhee and J. Cioffi, “On the capacity of multiuser wireless channels with
*multiple antennas,” IEEE Trans. on Info. Theory, vol. 49, no. 10, pp. 2580–*

2595, Oct. 2003.

[11] J. Chung, C. Hwang, K. Kim, and Y. Kim, “A random beamforming technique in MIMO systems exploiting multiuser diversity,” vol. 21, no. 5, pp. 848–855, June 2003.

[12] M. Sharif and B. Hassibi, “On the capacity of MIMO broadcast channel
*with partial side information,” IEEE Trans. on Info. Theory, vol. 51, no. 2,*
pp. 506–522, Feb. 2005.

[13] I. Kim, S. Hong, S. Ghassemzadeh, and V. Tarokh, “Optimum opportunistic
*beamforming based on multiple weighting vectors,” IEEE Trans. on Wireless*
*Communications, no. 6, pp. 2683–2687, Nov. 2005.*

[14] M. Kountouris and D. Gesbert, “Memory-based opportunistic multi-user
*beamforming,” in Proc., IEEE Intl. Symposium on Information Theory, Sept.*

2005, pp. 1426–1430.

[15] W. Choi, A. Forenza, J. G. Andrews, and R. W. Heath. Jr., “Opportunistic
*space division multiple access with beam selection,” to appear, IEEE Trans.*

*on Commun.*

[16] R. Gozali, R. M. Bueherer, and B. D. Woerner, “The impact of multiuser
*diversity on space-time block coding,” IEEE Communications Letters, vol. 7,*
pp. 213–215, May 2003.

[17] E. G. Larsson, “On the combination of spatial diversity and multiuser
*diversity,” IEEE Communications Letters, vol. 8, pp. 517–519, Aug. 2004.*

*[18] W. Choi et al., “The effects of co-channel interference on spatial diversity*
*techniques,” to appear, IEEE Wireless Communications and Networking*
*Conference, Mar. 2007.*

*[19] H. A. David and H. N. Nagaraja, Order Statistics, 3rd ed.* Wiley.

*[20] Effective-SNR Mapping for Modeling Frame Error Rates in Multiple-State*
*Channels, 3GPP2-C30-20030429-010 Std., Apr. 2003.*

*[21] cdma 2000 Evaluation Methodology, 3GPP2 C.P1002-C-0 Std., Sep. 2004.*

*[22] System level evaluation of OFDM- further considerations, Ericsson, TSG-*
RAN WG1 no.35, R1-03-1303 Std., Nov. 2003.

*[23] Effective SIR Computation for OFDM System-Level Simulations, Nortel,*
TSG-RAN WG1 no.35, R03-1370 Std., Nov. 2003.

*[24] OFDM Exponential Effective SIR Mapping Validation, EESM Simulation*
*Results for System-Level Performance Evaluations, 3GPP TSG-RAN1 Ad*
Hoc, R1-04-0089 Std., Jan. 2004.

*[25] PHY Abstractions for System Level Simulation, Qualcomm and Intel, IEEE*
802.11-04/0174r0 Std., 2004.

[26] L. Wan, S. Tsai, and M. Almgren, “A fading-insensitive performance metric
*for a unified link quality model,” in Proc., IEEE Wireless Communications*
*and Networking Conf., 2006.*

*[27] System-level evaluation of transmit diversity schemes with frequency-*
*dependent scheduling, Intel, TSG-RAN WG1 R1-061961 Std., 2006.*

−300 −20 −10 0 10 20 30 40

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

SINR (dB)

CDF

SISO (Mr=1, Mt=1) RD (Mr=2, Mt=1) STBC (Mr=1, Mt=2) STBC (Mr=2, Mt=2) CDD (Mr=1, Mt=2) CDD (Mr=2, Mt=2) Coherent BF (Mr=1, Mt=2) Monte Carlo simulations

SISO, 2x1 CDD 2x1 STBC

1x2 RD, 2x2 CDD, 2x1 BF 2x2 STBC

Fig. 1. CDF of post-processing SINRs where there are three (*K = 3) i.i.d.*

interferers with identical power*P**I*(*P*0*/P**I**= 7.5dB).*

−300 −20 −10 0 10 20 30 40

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

SINR (dB)

CDF

SISO (Mr=1, Mt=1) RD (Mr=2, Mt=1) STBC (Mr=1, Mt=2) STBC (Mr=2, Mt=2) CDD (Mr=1, Mt=2) CDD (Mr=2, Mt=2) Coherent BF (Mr=1, Mt=2)

2x1 STBC

SISO, 2x1 CDD

1x2 RD, 2x2 CDD, 2x1 BF 2x2 STBC

Fig. 2. CDF of maximum SINR when the number of users is 30 (*U = 30).*

Three (*K = 3) i.i.d. interferers with identical power P**I*(*P*0*/P**I**= 7.5dB) are*
considered.

−2 0 2 4 6 8 10 12 14

10^{−20}
10^{−18}
10^{−16}
10^{−14}
10^{−12}
10^{−10}
10^{−8}
10^{−6}
10^{−4}
10^{−2}
10^{0}

Avg. received SINR (dB)

Symbol Error Rate

SISO (Mr=1, Mt=1) RD (Mr=2, Mt=1) STBC (Mr=1, Mt=2) STBC (Mr=2, Mt=2) CDD (Mr=1, Mt=2) CDD (Mr=2, Mt=2) coherent BF (Mr=1, Mt=2)

1x2 RD, 2x2 CDD, 2x1 BF 2x2 STBC

2x1 STBC

SISO, 2x1 CDD

Fig. 3. Average symbol error rate (SER) for QPSK modulation versus average
received SINR when the number of users is 30 (*U = 30). Three (K = 3)*
i.i.d. interferers with identical power are considered. Average received SINR =
*P*0*/(3P**I*).

−6 −4 −2 0 2 4 6 8 10 12

1 2 3 4 5 6 7 8

Avg. received SINR (dB)

Spectral Efficiency (bps/Hz)

SISO (Mr=1, Mt=1) RD (Mr=2, Mt=1) STBC (Mr=1, Mt=2) STBC (Mr=2, Mt=2) CDD (Mr=1, Mt=2) CDD (Mr=2, Mt=2) Coherent BF (Mr=1, Mt=2)

2x2 STBC SISO, 2x1 CDD

1x2 RD, 2x2 CDD, 2x1 BF

2x1 STBC

Fig. 4. Spectral efficiency versus average received SINR when the number of
users is 30 (*U = 30). Three (K = 3) i.i.d. interferers with identical power are*
considered. Average received SINR =*P*0*/(3P**I*).

0 5 10 15 20 25 30

1 1.5 2 2.5 3 3.5 4

Number of users, U

Spectral Efficiency (bps/Hz)

SISO (Mr=1, Mt=1) RD (Mr=2, Mt=1) STBC (Mr=1, Mt=2) STBC (Mr=2, Mt=2)) CDD (Mr=1, Mt=2) CDD (Mr=2, Mt=2) Coherent BF (Mr=1, Mt=2)

2x1 STBC SISO, 2x1 CDD

1x2 RD, 2x2 CDD, 2x1 BF

2x2 STBC

Fig. 5. Spectral efficiency versus average received SINR versus the number
of users. Three (*K = 3) i.i.d. interferers with identical power are considered.*

Average received SINR =*P*0*/(3P**I*).

TABLE I

SUMMARY OFANALYTICALRESULTS(*M**r**× M**t*,*K*INTERFERERS WITH IDENTICAL POWER*P**I*,DESIRED POWER*P*0)

*E[γ]* *Var[γ]* *f**γ**(γ)*

SISO *P*0

*P**I**(K − 1)*

*P*_{0}^{2}
*P*_{I}^{2}

*K*
*(K − 1)*^{2}*(K − 2)*

*P**I*

*P*0

*Γ(1 + K)*
*Γ(1)Γ(K)*

1

1 +^{P}_{P}^{I}_{0}*γ*_{1+K}

RD (Mt= 1) *P*0*M**r*

*P**I**(K − 1)*

*P*_{0}^{2}
*P*_{I}^{2}

*M**r**(M**r**+ K − 1)*
*(K − 1)*^{2}*(K − 2)*

*P**I*

*P*0

*Γ(M**r**+ K)*
*Γ(M**r**)Γ(K)*

(^{P}_{P}^{I}_{0}* ^{γ}*)

^{M}

^{r}

^{−1}

1 +^{P}_{P}^{I}_{0}^{γ}_{M}_{r}_{+K}

STBC *P*0*M**r**M**t*

*P**I**(KM**t**− 1)*
*P*_{0}^{2}
*P*_{I}^{2}

*M**r**M**t**(M**r**M**t**+ KM**t**− 1)*
*(KM**t**− 1)*^{2}*(KM**t**− 2)*

*P**I*

*P*0

*Γ(M**r**M**t**+ KM**t*)
*Γ(M**r**M**t**)Γ(KM**t*)

(^{P}_{P}^{I}_{0}* ^{γ}*)

^{M}

^{r}

^{M}

^{t}

^{−1}

1 +^{P}_{P}^{I}_{0}^{γ}_{M}_{r}_{M}_{t}_{+KM}_{t}

CDD *P*0*M**r*

*P**I**(K − 1)*

*P*_{0}^{2}
*P*_{I}^{2}

*M**r**(M**r**+ K − 1)*
*(K − 1)*^{2}*(K − 2)*

*P**I*

*P*0

*Γ(M**r**+ K)*
*Γ(M**r**)Γ(K)*

(^{P}_{P}^{I}_{0}* ^{γ}*)

^{M}

^{r}

^{−1}

1 +^{P}_{P}^{I}_{0}^{γ}_{M}_{r}* _{+K}*
Coherent BF (Mr= 1)

*P*0

*M*

*t*

*P**I**(K − 1)*

*P*_{0}^{2}
*P*_{I}^{2}

*M**t**(M**t**+ K − 1)*
*(K − 1)*^{2}*(K − 2)*

*P**I*

*P*0

*Γ(M**t**+ K)*
*Γ(M**t**)Γ(K)*

(^{P}_{P}^{I}_{0}* ^{γ}*)

^{M}

^{t}

^{−1}
1 +^{P}_{P}^{I}_{0}^{γ}

_{M}_{t}_{+K}

−300 −20 −10 0 10 20 30 40

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

SINR (dB)

CDF

RD (Mr=2, Mt=1) STBC (Mr=2, Mt=2) CDD (Mr=2, Mt=2)

−300 −20 −10 0 10 20 30 40

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

SINR (dB)

CDF

RD (Mr=2, Mt=1) STBC (Mr=2, Mt=2) CDD (Mr=2, Mt=2)

2x2 STBC RD, 2x2 CDD

Fig. 6. CDF of EESM for all resource blocks in flat fading channels before frequency selective opportunistic scheduling.

−300 −20 −10 0 10 20 30 40

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

SINR (dB)

CDF

RD (Mr=2, Mt=1) STBC (Mr=2, Mt=2) CDD (Mr=2, Mt=2)

2x2 CDD 2x2 STBC

RD

Fig. 7. CDF of EESM in flat fading channels after frequency selective opportunistic scheduling.

TABLE II

*2 × 2*ANTENNA CONFIGURATION WITH*U = 1 (K = 3*^{AND}*P**s**/P**I**= 7.5dB)*
*E[γ] in dB* *Var[γ] in dB*

SISO (reference) *4.47* *13.52*

*1 × 2 RD* *7.48* *17.78*

*2 × 1 STBC* *3.53* *6.47*

*2 × 2 STBC* *6.51* *10.52*

*2 × 1 CDD* *4.47* *13.61*

*2 × 2 CDD* *7.51* *17.70*

*2 × 1 BF* *7.51* *17.63*

TABLE III

*2 × 2*ANTENNA CONFIGURATION WITH*U = 30 (K = 3*^{AND}
*P**s**/P**I**= 7.5dB)*

E[max

*U* *γ] (dB)*

Simulation Analy. upper bound

Var[max*U**γ] (dB)*

SISO 12.6 13.1 *23.5*

*1 × 2 RD* 14.8 15.4 28

*2 × 1 STBC* 9.5 10.1 12

*2 × 2 STBC* 11.7 12.3 *15.5*

*2 × 1 CDD* 12.6 13.1 *23.7*

*2 × 2 CDD* 14.8 15.4 28

*2 × 1 BF* 14.8 15.4 28